Linearizing Symulink dynamic model to retrieve coefficient matrices A B C D (without using linmod function) with Least Squares method

Hello,

I wonder how to implement linearazing the dynamic model from simulink to the linear one. I know that there are built-in functions, however, I have to implement this from scratch using Least Squares method.

The model consists of two fluid tanks with free drainage. The maximum height of tank is 15m, which I have already done in Simulink.

Subsystem

I have to use h1, h2, h1_p (derivate of h1), h2_p (derivate of h2), and "u", which is my Qwe (input signal 5.373)

Could you help me to implement the "Least Squares" method to retrieve A, B, C, D matrices that are part of the linear model?

My start program:

[Sam Chak]

Hi Adrian,

Could you list out the differential equations for the two fluid tanks with parameters?

Are you allowed to use the linearize function?

[Adrian Tworek]

Hi Sam,

here are the differential equations:

Everything works perfectly in Simulink. Initial condition in both tanks integrators is 0.

The main purpose is to retrieve linear matrices (A, B, C, D) without using any built-in functions that return them immediately in one line (like linmod etc.). It has to be calculated in matlab file using Least Square method and h1, h2, h1_derivative, h2_derivate and input signal (Qwe).

Thank you for help!

[Sam Chak]

Hi @Adrian Tworek

Thanks for the ODEs. I know that least squares method is commonly used in curve-fitting problems, dealing with sparse matrices, and data-driven system identification. But the linearization method that I know of usually involves finding the Jacobian.

Perhaps you can first create the symbolic differential equations. Follow the examples here.

https://www.mathworks.com/help/symbolic/analyze-and-manipulate-differential-algebraic-equations.html

https://www.mathworks.com/help/symbolic/solve-a-system-of-differential-equations.html

syms h1(t) h2(t) Q S1 S2 Sw1 Sw2 phi1 phi2 g

ode1 = diff(h1) == Q/S1 - (Sw1*phi1*sqrt(2*g*(h1 - h2)))/S1;
ode2 = diff(h2) == (Sw1*phi1*sqrt(2*g*(h1 - h2)))/S2 - (Sw2*phi2*sqrt(2*g*h2))/S2;
odes = [ode1; ode2]

Let's hope other users who are good at math/physics/MATLAB can help out.